Probability+B-1

=__ Probability __= toc Probability is the ratio (fraction or percent or decimal) of the number of events you want to have happen, divided by the number of possible events. Factorial is the product when a positive integer, whole numbers, is multiplied by all the positive integers between the integer and 1.

Example: What is the factorial of 4? 4! = 4*3*2*1 4! = 24

=__ Permutations __=

If you have a set of objects, a permutation is the number of different ways you can arrange the objects. The order ** does **matter. Example: [1,2,3] = A set of 3 numbers. The 3 numbers can be arranged in any order. One way to find out the order is to write it out. [1,2,3] [1,3,2] [2,1,3] [2,3,1] [3,2,1] [3,1,2]

There are 6 permutations. Another way to find out how many ways to arrange the set is to use factorial. Since there are 3 numbers in each set, you use 3! which comes out as

( 3 numbers to pick from) *(2 numbers to pick from) *1 (number left to choose) = 6 permutations.

If there are 'n' objects, and you want to pick out 'r' objects then the formula of the number of permutations is

math \frac{n!}{(n-r)!} math

Example 2: How many ways are there of picking 2 numbers from the set {1,2,3} if the order does matter?

Solution:

math \frac{3!}{(3-2)!} = \frac{3!}{1!} = 6 math

Therefore, there are 6 ways of picking 2 numbers out of a set of 3, where the order does matter. The six ways are: [1,2], [2,1] , [1,3] , [3,1], [2,3] ,and [3,2].

= = =__ Combinations __=

A combination is the number of ways of selecting objects when the order in which you select them does not matter. Example #1: The arrangement of 1,2,3 is the same as the arrangement 3,2,1.

The formula for selecting a combination of 'r' objects from a set of 'n' objects is....... math \frac{n!}{(n-r)!r!} math

Example #2: How many ways are there for choosing 2 numbers from the set {1,2,3}, if the order does not matter?

Solution: The numbers of ways of picking a combination of 2 numbers from the set {1,2.3} is given by because there are 3! = 6 ways of arranging the object, but

math \frac{1}{(3-2)!2!} = \frac{1}{1!2!} = \frac{1}{2} math

Only 1/2 of those 6 arrangements are unique. The others are repetitions. [1,2] is the same as [2,1] [1,3] is the same as [3,1] [2,3] is the same as [3,2]

Therefore, there are 6/2 = 3 unique ways of picking 2 numbers from the set {1,2,3}, where the order does not matter.

__ Further Discussion __

 * What are real world applications of the use of combinations?
 * What re some real world applications of the use of permutations?